1.

Prove that : (sin(90 - A))/(1 - tan A) + (sin A)/(1 - tan(90 - A)) = cos A + cos(90 - A) .​

Answer» <html><body><p>-step explanation:S=sin(−660)tan(1050)sec(−420)(cos225)(cosec315)(cos510) We make the following changes:sec x=1/cosxSec(-420) = [(1/cos(-420)]cosec x =1 / sin xCosec(315) = [(1/sin(315)]<a href="https://interviewquestions.tuteehub.com/tag/thus-2307358" style="font-weight:bold;" target="_blank" title="Click to know more about THUS">THUS</a>, S=sin(−660)tan(1050)sin(315)cos(225)cos(−420)cos(510)Now,sin (-x) = - sin xsin (-660) = - sin 660Cos (-x) = cos xCos (-420) = cos 420Thus, S=−sin(660)tan(1050)sin(315)cos(225)cos(420)cos(510)We further <a href="https://interviewquestions.tuteehub.com/tag/use-1441041" style="font-weight:bold;" target="_blank" title="Click to know more about USE">USE</a> the identities:Sin (2π + x ) = sin xSin (660) = sin ( <a href="https://interviewquestions.tuteehub.com/tag/360-309472" style="font-weight:bold;" target="_blank" title="Click to know more about 360">360</a> + 300) = sin ( 2π + 300) = sin (300)Sin ( π + x) = - sin xSin (300) = sin ( 180 + 120) = sin (π +120) = -(-sin 120 )Sin (120) = sin ( 180 - 60) = sin 60- sin(660) = sin (60)Similarly, Sin (315) = - sin (45)tan x = sin x / cos xtan (1050) = sin (1050)/ cos (1050) = sin (<a href="https://interviewquestions.tuteehub.com/tag/2-283658" style="font-weight:bold;" target="_blank" title="Click to know more about 2">2</a>*2π + 330)/cos (2*2π + 330) = sin (330)/cos(330)Sin ( 330) = sin ( 180 + <a href="https://interviewquestions.tuteehub.com/tag/150-275254" style="font-weight:bold;" target="_blank" title="Click to know more about 150">150</a>) = sin (π +150) = (-sin 150 )Sin (150) = sin ( 180 - 30) = - sin 30Cos( 330) = cos( 180 + 150) = cos(π +150) = (-cos150 )Cos (150) = cos( 180 - 30) =-(-cos 30) = cos 30Tan (1050) = tan (30)Cos(2π + x ) = cos xCos(420) =cos (360 +60) = cos (2π +60) =cos 60Cos (420) =cos (60)Cos (510) =cos (360 +150) = cos (2π +150) = cos 150Cos(π-x) = -cos xCos ( 150) =cos (180 -30) = cos (π - 30) = - cos (30)Cos (510) = - cos (30)Cos ( π + x) = - cos xCos (225) = cos ( 180 + 45) = cos (π +45) = - cos (45)Cos (225) = - cos 45Thus, S=−sin(660)tan(1050)sin(315)cos(225)cos(420)cos(510)All the values fall in quadrant 1. So all are positive.S=sin(60)tan(30)sin(45)cos(45)cos(60)cos(30)S=(√3/2)∗(1/√3)∗(1/√2)(1/√2)∗(1/2)∗(√3/2)S=2/√3</p></body></html>


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